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Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy
Author(s) -
Konopelchenko B. G.,
Landolfi G.
Publication year - 2000
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00133
Subject(s) - integrable system , mathematics , euclidean geometry , surface (topology) , representation (politics) , mean curvature , pure mathematics , curvature , hierarchy , mathematical analysis , weierstrass functions , willmore energy , geometry , mean curvature flow , politics , political science , economics , law , market economy
Extensions of the generalized Weierstrass representation to generic surfaces in 4‐D Euclidean and pseudo‐Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey–Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.